Optimization Letters

, Volume 2, Issue 2, pp 143–155 | Cite as

Slopes of shadow prices and Lagrange multipliers

  • S. D. Flåm
  • H. Th. Jongen
  • O. Stein
Original Paper


Many economic models and optimization problems generate (endogenous) shadow prices—alias dual variables or Lagrange multipliers. Frequently the “slopes” of resulting price curves—that is, multiplier derivatives—are of great interest. These objects relate to the Jacobian of the optimality conditions. That particular matrix often has block structure. So, we derive explicit formulas for the inverse of such matrices and, as a consequence, for the multiplier derivatives.


Sensitivity Optimal value function Shadow price Karush-Kuhn-Tucker system Matrix inversion 


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Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Department of EconomicsUniversity of BergenBergenNorway
  2. 2.Department of Mathematics-CRWTH Aachen UniversityAachenGermany
  3. 3.Department of EconomicsUniversity of KarlsruheKarlsruheGermany

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